Diagonalization of transfer matrix of supersymmetry $u_q(hat{sl}(m+1|n+1))$ chain with a boundary

We study the supersymmetry $U_q(\hat{sl}(M+1|N+1))$ analogue of the supersymmetric t-J model with a boundary, in the framework of the algebraic analysis method. We diagonalize the commuting transfer matrix by using the bosonization of the vertex operator associated with the quantum affine supersymmetry.

💡 Research Summary

The paper investigates a boundary version of the supersymmetric t‑J model whose underlying symmetry is the quantum affine superalgebra (U_q(\widehat{sl}(M+1|N+1))). The authors adopt the algebraic‑analysis framework, which combines the quantum inverse scattering method with a bosonized construction of vertex operators. First, they define the bulk model in terms of the (R)-matrix of the superalgebra and introduce a boundary (K)-matrix that solves the reflection equation. The chosen (K)-matrix respects the co‑module structure of the superalgebra and depends on two boundary parameters that control the strength of the interaction at the edge.

A central technical achievement is the explicit bosonization of the type‑I and type‑II vertex operators associated with (U_q(\widehat{sl}(M+1|N+1))). By introducing free bosonic fields for the Cartan subalgebra and free fermionic fields for the odd roots, the authors realize the superalgebra on a Fock space. The vertex operators are written as normal‑ordered exponentials of these fields, and their (q)-deformed commutation relations are derived directly from the free‑field OPEs. This construction makes it possible to write the double‑row transfer matrix (the boundary transfer matrix) as a product \